Why PSA Recurrence Needs Careful Definition
After radical prostatectomy, prostate-specific antigen (PSA) is monitored because PSA should generally fall to a very low level when prostate tissue has been removed. A subsequent rise can be evidence of biochemical recurrence, but the clinical interpretation depends on how the PSA was measured, the assay’s lower reporting limit, and the study’s definition of recurrence. These details affect both the longitudinal trajectory and the event-time data supplied to a joint model.
A commonly used biochemical-recurrence convention after prostatectomy is PSA at or above 0.2 ng/mL, confirmed by a subsequent measurement at or above 0.2 ng/mL. Definitions and event-dating conventions vary across protocols. A study might date the event at the first qualifying value, even though confirmation is only known later, or at the confirmation date itself. Those are different endpoints. The rule and date convention must be stated before analyzing the data.
Joint modeling can relate an individual’s underlying PSA trajectory to a distinct time-to-event outcome. A crucial distinction in this case study is whether the event is genuinely distinct from the PSA readings, or whether the event is calculated from those very readings. The familiar product of a longitudinal likelihood and a survival likelihood requires conditional independence that may not hold in the second situation.
Representing Low and Unquantifiable PSA Results
A laboratory may report a value as “less than 0.05 ng/mL” rather than provide a numerical measurement. This is information about the PSA: the value is below the assay limit, not known to be zero or equal to half the limit. Substituting zero or 0.025 ng/mL creates an artificial point observation and can distort the estimated postoperative trajectory, especially when many results are near the limit.
For illustration, suppose the longitudinal model for PSA on its original scale is \(Y_{ij}=m_i(t_{ij})+\epsilon_{ij}\), with \(\epsilon_{ij}\sim N(0,\sigma^2)\) conditional on the subject-specific trajectory. Here \(m_i(t)\) is the modeled mean PSA at time \(t\). If the laboratory reports only that \(Y_{ij}<L\), the likelihood contribution is the probability of being below \(L\), not a density evaluated at an invented value:
This expression assumes the stated normal measurement-error model is suitable on the chosen scale and that the reporting limit is known. If a log transformation is used, zero and below-limit results need an appropriate censored-data treatment on that scale; one cannot take the logarithm of zero. The scale, residual distribution, assay limits, and any changes in assay over calendar time should be checked against the actual data and laboratory documentation.
Worked Example: Using a PSA Assay Limit
Suppose a hypothetical postoperative sample is reported as PSA <0.05 ng/mL. At this visit, the fitted trajectory has mean \(m_i(t_{ij})=0.032\) ng/mL, and the assumed residual standard deviation is \(\sigma=0.020\) ng/mL. Under the conditional normal model, its contribution is
Thus the observed report contributes a probability of about 0.816 under these model values. It does not say that the true PSA was 0.025 ng/mL, nor that the measurement error was exactly the difference between 0.025 and 0.032. The computation assumes the assay limit and the specified normal error model apply; a substantial pattern of residual or censoring-model misfit would call for reconsidering those assumptions.
Applying a Confirmed-Threshold Rule
The recurrence rule should be applied to the sequence of available results, not to a smoothed model curve as if that curve were the observed endpoint. The fitted trajectory estimates an underlying pattern; the protocol’s endpoint is assigned from measurements according to its stated threshold and confirmation rule. In particular, one elevated value alone does not satisfy a rule requiring two consecutive qualifying values.
Worked Example: First Crossing and Confirmation
Consider hypothetical PSA measurements after surgery, with the recurrence rule defined as two consecutive readings at or above 0.20 ng/mL. Measurements are 0.08 ng/mL at 6 months, 0.13 ng/mL at 12 months, 0.21 ng/mL at 18 months, and 0.24 ng/mL at 21 months.
The 6- and 12-month results are below threshold. The 18-month result is the first value at or above 0.20, but it is not confirmed by a qualifying second value yet. The 21-month result is also at or above 0.20, so the two-consecutive-reading rule is met at that point.
If the protocol dates recurrence at the confirmation visit, the operational event time is 21 months. If it dates recurrence at the first qualifying reading, retrospectively confirmed, it assigns 18 months once the 21-month result is available. Under the latter interpretation, the crossing is known to have occurred after the 12-month below-threshold reading and by the 18-month qualifying reading, so its observation brackets the crossing in \((12,18]\) months. These conventions must not be interchanged silently: they yield different event-time data and can affect estimated associations.
This example assumes the four measurements are the relevant consecutive measurements under the protocol and that there are no intervening qualifying results. If visits are missed, values are missing, or the rule specifies a different confirmation window, the endpoint must follow that actual rule rather than an informal visual judgment.
When the Standard Joint Likelihood Applies
Let \(b_i\) denote subject \(i\)’s random effects, \(Y_i\) the longitudinal observations, \(T_i\) the event or censoring time, and \(\delta_i\) the event indicator, with \(\delta_i=1\) for an observed event and \(\delta_i=0\) for right censoring. A standard shared-random-effects likelihood can use a longitudinal contribution and an event contribution when the event process is distinct from the PSA readings in the longitudinal likelihood, and when the two are conditionally independent given \(b_i\) and included covariates. The observation process must also be adequately handled—for example, visit timing cannot be assumed harmless if it is driven by unmodeled health changes related to the event.
Under those assumptions, if the event hazard conditional on \(b_i\) is \(h_i(t\mid b_i)\), with conditional survival function \(S_i(t\mid b_i)\), the event contribution is \(h_i(T_i\mid b_i)^{\delta_i}S_i(T_i\mid b_i)\). The subject-level contribution integrates the product over the random-effects distribution:
This is not a universal formula for every endpoint called “recurrence.” It is appropriate only if the stated conditional-independence and observation-process assumptions are credible for the particular endpoint and recorded data. The fact that a PSA trajectory is included in the hazard model does not by itself establish that independence.
For example, one might model a separate, clinically adjudicated event time using a PSA trajectory in the hazard, such as \(h_i(t\mid b_i)=h_0(t)\exp\{\gamma^\mathsf{T}X_i+\alpha m_i(t)\}\). Here \(h_0(t)\) is the baseline hazard, \(X_i\) contains covariates, and \(\alpha\) describes the conditional association between the current modeled PSA level and the event hazard. The formula is an illustrative association structure, not a claim that any particular recurrence endpoint is independent of PSA measurements.
Worked Example: Interpreting a PSA–Hazard Association
Suppose, hypothetically, a fitted model for a separate adjudicated clinical event has a current-value association estimate \(\alpha=2.0\) per ng/mL. Compare two otherwise identical subjects at a time when their modeled PSA values are 0.10 and 0.20 ng/mL. Their PSA difference is 0.10 ng/mL, so the conditional hazard ratio is
Under this model, at that time and conditional on the same covariates and random-effects specification, the hazard is about 1.221 times as large for a 0.10 ng/mL higher modeled PSA value. The units of \(\alpha\) are inverse ng/mL. This is an association in the specified model, not evidence that increasing PSA causes the event. In practice the scale and range of PSA values matter; extrapolating this log-linear association beyond the observed range may be unjustified.
This calculation presumes the event is distinct from the PSA measurements used in the longitudinal likelihood and that the joint-model assumptions are defensible. It should not be reported as the hazard ratio for a recurrence indicator mechanically constructed from those same readings without addressing their dependence.
Why a PSA-Derived Event Cannot Simply Be Multiplied In
When biochemical recurrence is defined directly from the same PSA values included in \(Y_i\), event status and event timing are functions of those values. Conditional on the values, the protocol rule determines the event label. The event contribution therefore is not conditionally independent of those observations given \(b_i\). Multiplying \(f(Y_i\mid b_i)\) by a separate survival contribution for the rule-derived event would generally count information from the PSA readings twice.
The dependence can be seen without a complex model. Let \(D=1\) when a single hypothetical reading \(Y\) exceeds a threshold \(c\), and \(D=0\) otherwise. Once the reading is observed, \(D\) is determined. Its joint contribution with the reading is
where \(\mathbf{1}\{\cdot\}\) equals 1 when the recorded event label agrees with the rule and 0 otherwise. A product \(f(y\mid b)P(D\mid b)\) adds the marginal probability of an event given \(b\) even though the observed reading has already determined \(D\). For a confirmed-threshold rule the logic is the same, but the deterministic rule uses a sequence of readings, their order, and potentially their visit times.
One alternative is to treat the recurrence label and date as derived summaries of the measurement record and model the measurement process, visit times, and deterministic rule coherently. Another is to define a genuinely distinct event, such as a separately adjudicated clinical outcome, and assess whether a joint model with PSA is appropriate. A model for an unobserved biological recurrence time is also possible in principle, but it must specify how that latent event relates to the PSA measurements and how the observed threshold rule provides information about it. Merely relabeling the rule-based date as a latent event time does not solve the dependence problem.
Diagnostics, Alternatives, and Limitations
For a PSA analysis, start by auditing the measurement record and endpoint definition. Check whether reported values are exact measurements or categories such as “below limit,” whether assay limits changed, and whether units are consistent. Verify consecutive-reading logic, missing or irregular visits, and the event-date convention against the protocol. A recurrence date earlier than its confirmation may be an intentional retrospective date, not a recording error, but it must be interpreted accordingly.
For a continuous longitudinal model, examine whether the chosen trajectory and residual assumptions capture the observed pattern, particularly near the assay floor. Unexpected piles of identical low values may indicate that censored reports were converted to numbers. For an appropriate distinct event process, assess whether the selected hazard association and baseline hazard are plausible and whether the fit is stable; software warnings do not establish that the model is suitable. If visits become more frequent as PSA rises, a model that ignores the visit process may not represent the data-collection mechanism adequately.
| Approach | When it may be suitable | Important caution |
|---|---|---|
| Joint model with a distinct event outcome | The event is not calculated from the PSA observations in the longitudinal likelihood, and conditional independence given modeled effects and covariates is defensible. | Confirm the endpoint and visit-process assumptions; the model does not make a PSA-derived event independent. |
| Model PSA observations and the rule-derived endpoint together | The event is assigned from the same serial PSA record, and the measurement process and recurrence rule can be represented. | The rule, timing, assay censoring, and potentially informative measurement schedule need explicit treatment. |
| Analyze the PSA trajectory and recurrence rule separately | Separate descriptive or protocol-specific questions are the goal. | Separate analyses do not automatically provide a coherent joint likelihood or correct dependence-adjusted inference. |
No single strategy removes all limitations. A particular recurrence threshold may not capture every clinically meaningful pattern, and the assay floor can limit what is learned about small postoperative differences. Event dating can be intrinsically coarse because PSA is observed only at visits. Model-based trajectories add assumptions about shape and measurement error; hazard associations depend on the chosen event definition and model. Conclusions should therefore name the assay, rule, timing convention, and outcome actually analyzed.
Common Mistake
- Wrong reasoning: “I have a PSA likelihood and a recurrence time, so I can multiply the longitudinal density by a survival contribution conditional on the random effects.”
- Correction: First determine whether the recurrence time and indicator were calculated from those same PSA values. If they were, the endpoint is dependent on the observations by construction. Do not use the standard product as their joint likelihood without modeling that dependence. The product is available only when the event process is distinct and the conditional-independence and observation assumptions are justified.
Check Your Understanding
Answer each question using the stated assumptions. Brief solutions follow.
- A laboratory reports PSA <0.10 ng/mL. Why is replacing it with zero not generally justified?
- Under a two-consecutive-values rule at 0.20 ng/mL, readings are 0.19, 0.22, and 0.18 ng/mL. Has recurrence been confirmed?
- A recurrence indicator is calculated from the same PSA sequence used in the longitudinal model. May the standard independent event contribution simply be multiplied into the longitudinal likelihood? Explain.
- A separate event model gives \(\alpha=1.5\) per ng/mL for current modeled PSA. What is the conditional hazard ratio for a 0.20 ng/mL higher modeled PSA value?
- If a protocol dates recurrence at the first threshold crossing but requires later confirmation, what does the later reading establish about the first-crossing date?
Solutions. (1) The report gives an interval, PSA below 0.10, not a known point value; zero substitution invents a measurement and may distort the trajectory. (2) No. Only the 0.22 reading qualifies; the next value is below threshold, so there are not two consecutive qualifying readings. (3) No, not without modeling the dependence. The rule-derived event is a function of those measurements, so multiplying a separate survival contribution may count their information twice. (4) The ratio is \(\exp(1.5\times0.20)=\exp(0.30)\approx1.350\), conditional on the model and other covariates. (5) Confirmation establishes retrospectively that the first qualifying reading met the protocol’s confirmed rule; the assigned date may be that first reading, even though the rule could not be verified until the later measurement.